The Characteristic Value Problem—Generalities
John A. Todd · Birkhäuser Basel eBooks · 1977
It has been said that the main problem of applied mathematics is to find the characteristic values and vectors of symmetric matrices, or, what is the same thing, the principal axes of ellipsoids. In some cases we only require crude estimates for the characteristic values, in others we require close approximations to a single characteristic value — very often the dominant one — or, may be, to a few, while sometimes we may require all the characteristic values and vectors. The problem becomes very much more difficult if the matrix is not symmetric — in the first place, the characteristic values are no longer necessarily real. We also encounter the so-called “generalized characteristic value problem”, the determination of non-trivial solutions of $$Ax = \lambda Bx.$$